1. Field Of The Invention
This invention relates to an improved double-diffractometer for the precise measurement of lattice parameters in single crystal wafers, and more particularly to a technique for eliminating tilt misalignment between two independent single crystals where one crystal is misoriented in an arbitrary direction with respect to the other crystal, prior to comparing their diffraction angles.
2. Background Description
The precise measurement of lattice parameters in single crystals is important in at least two areas of research. In electronic applications, this information is used in fabricating epitaxial thin films of Ga.sub.1-x Al.sub.x As on GaAs as well as Ga.sub.1-x Al.sub.x As on Ga.sub.1-7 In.sub.4 As, and In.sub.x Ga.sub.1-x As on InP. These formations require exact knowledge of the substrate lattice parameter, since often a comparison with the substrate yields an accurate measure of the epilayer lattice parameter and hence a measure of the alloying content. The control of alloying content is essential in preparing lattice-matched layers and thin films useful for proper operation of microelectronic devices. In materials science, on the other hand, information on lattice parameters is used for investigating the mechanical and optical properties of materials and their variation as an indicator of the defect structure and strain. A simple measurement method, sufficiently sensitive to small variation in the lattice parameter would therefore be desirable.
During the last three decades several techniques using x-ray diffraction have been introduced which provide accuracies varying from one one-hundredth of an angstrom (0.01.ANG.) to 0.00001.ANG.. Most recent applications of such measurements have required the upper levels of resolution, i.e., better than .001.ANG., where the effects of strain and doping can be quantified as a function of minute changes in the lattice parameter. The most well known of these is the Bond method, as described in Acta Cryst., 13 (1960) 814, by W. L. Bond, which permits absolute measurement of diffraction angles from which the lattice parameter may be calculated. This technique, while in principle is unsurpassed for its accuracy has nevertheless several drawbacks, one of which is the complexity and the bulk of the instrumentation which relies on a very accurate and therefore expensive theta circle for positioning the crystal. The second is the relatively long measurement time and the difficulties associated with automation which reduce this time to only about 30 minutes per measurement. The third drawback is that to attain the part-per-million accuracy the samples must be perfect over a relatively wide area. Although this last condition is necessary for all high resolution measurements, it is not a condition which can be satisfied for most crystals of interest under development today. Thus, the state of perfection of most present-day crystals, except for Si and Ge, is such that the resolution required for their evaluation need not exceed a few ten parts per million. Rather, changes in the lattice constant corresponding to the fourth decimal place (.about.2.times.10.sup.-4 .ANG.) are sought. Such variations correspond to accuracies of the order of a second of arc in the Bragg angle .theta.. As a result, the Bond method remains an "umpire" method with principal applications in studies involving a limited number of crystals such as Si and Ge.
In addition to the Bond method, several other techniques have been introduced, all of which require specialized instrumentation without wide applicability to different materials. For example, multiple detectors, multiple sources, or specifically shaped reference crystals are used, all of which limit the application to the measurement of the variation of the lattice parameter rather than the actual parameter itself.
FIG. 1 is a perspective view of the basic structure of a typical double crystal diffractometer, such as Blake's, having two crystal stages 10 and 12. The first crystal 10 and the second crystal 12 are the main components and each of them is very accurately aligned. X-rays come from an x-ray source 14 and become partially collimated by slits S.sub.1 and S.sub.2 and strike the first crystal 10. Slit S.sub.2 is positioned between the second crystal 12 and a detector 15. Slit S.sub.2 is sometimes used in place of slit S.sub.2 . Both slits simply limit the beam so that it doesn't scatter. However, it is preferred to have both slit S.sub.1 and slit S.sub.2 near the x-ray source so that right away the total radiation is reduced to reduce the hazard of exposure. The crystals 10 and 12 have tilts .delta..sub.1 and .delta..sub.2 respectively between their normals n1 and n2 and a the horizontal plane, and the slant angle between the line connecting the slits S.sub.1 and S.sub.2 and the horizontal plane which create errors in the measurement of lattice parameters. Here the first crystal 10 is adjusted so that it can pick out diffraction angles near the characteristic radiation of the x-ray source 14--and then a characteristic radiation is reflected toward the second crystal 12. The second crystal 12 is the one which would ordinarily be in a usual double crystal diffractometer being tested. HoWever in the method proposed below in the preferred embodiment, two independent crystals take the place of the second crystal 12, i.e. a reference and an unknown crystal--both mounted in the same general area. One is translated into the x-ray beam and then the other one is translated into the beam while looking at their different diffraction angles. And by measuring the difference in diffraction angles of these two independent crystals one can calculate, by Bragg's law, the lattice parameter of the unknown crystal.
In what follows, we will first review the method as applied to a system of two independent crystals each having only a negligible component of vertical tilt. The effects of second crystal tilt and other geometrical constraints are then discussed.
Consider two crystals, C1 and C2 (unknown and standard), mounted together on the second stage of a double crystal diffractometer as shown in FIGS. 2A and 2B. The diffracting planes are assumed to be vertical, i.e., their normals, having a small misorientation angle o between them, are directed in the horizontal plane. Monochromatic X-rays arriving from the first crystal are diffracted sequentially from crystal C1 and crystal C2 by rotating the second stage. FIGS. 2A and 2B is a sketch of such an experiment in which, for the sake of simplicity, the X-ray beam--rather than the stage--is rotated. The separation between the two X-ray beams is an angle X containing both the difference in the Bragg angles .THETA..sub.B1 -.THETA..sub.B2 =.THETA..sub.B, and a misorientation angle .alpha.. In FIG. 2A, crystal C1 is shown closer to the X-ray source than crystal C2. The relationship between X, .alpha., and .THETA. is thus EQU X.sub.a =.DELTA..THETA..sub.B +.alpha.. (1)
If now the entire assembly were rotated about a horizontal azimuth axis by 180.degree., as in FIG. 2B, the roles of crystal C1 and crystal C2 would be interchanged, and a new quantity X.sub.b would be measured such that EQU X.sub.b =.DELTA..theta..sub.B -.alpha.. (2)
For both FIGS. 2A and 2B, angle X is the separation between two relatively sharp diffraction peaks resulting from the "parallel" geometry. Thus both .THETA. and .alpha. can be calculated from (1) and (2): EQU .DELTA..THETA.=(X.sub.a +X.sub.b)/2, (3) EQU .alpha.=(X.sub.a -X.sub.b)/2. (4)
Relations (3) and (4) are valid for all configurations of independent crystals C1 and C2 with negligible vertical tilt, i.e. their normals lie in the horizontal plane, provided the algebraic signs of angles X.sub.a and X.sub.b are also taken into account.
Although these relations were introduced more than two decades ago, no applications of the concept to two independent crystals, where one crystal is misoriented in an arbitrary direction with respect to the other, have been reported. The absence of published work in this area is probably due to the fact that the double crystal diffractometer is commonly thought to be suitable only for relative angular measurements on "coherent" epitaxial growths and those related to small grain boundary strains in one crystal alone. In these cases, most uncertainties associated with the double crystal alignment would nearly vanish due to the cancellation of instrumental factors common between two parts of the crystal, especially if adjustments other than the angle .THETA. were unnecessary. On the other hand, if a significant readjustment of the diffractometer were needed, such as in transition between two independently mounted crystals, the simple relationship among the parameters of eqs. (3) and (4) would no longer hold.
Mathematical analyses and supporting experiments reported by various workers show that aside from influencing the breadth of the rocking curve, the non-zero relative tilt between the first and second crystal, C1 and C2, appears as a measurable shift in the diffraction angle .THETA.. Since it is not unusual for any given crystal to be generally misaligned by 1.degree.-2.degree., an experimental difficulty arises in that as one crystal is tilted into the optimum diffracting position, the other may be tilted out, and vice versa. Presumably, then, one approach to removing the errors would be to alternately correct the tilt of each wafer before measuring its rocking curve. The procedure, however, would prove unsatisfactory due to the fact that any intermediate disturbance of the tilt mechanism would introduce significant errors (typically a few seconds of arc) in the ultimate difference .DELTA..THETA..sub.B. In short, sequential adjustment of the tilt for two wafers does not provide a highly accurate means of comparing their diffraction angles. The difficulty in comparing two arbitrarily mounted crystals arises from the fact that the measured diffraction angle .THETA. is a combination of the Bragg angle .THETA..sub.B, geometrical and instrumental parameters and errors of tilt and misalignment. For an analysis of the errors in measurement of the diffraction angles due to the tilt, See "A High-Resolution Double-Crystal Diffractometer Method For The Measurement Of Lattice parameter In Single Crystals", J. of Crystal Growth, 96 (1989) 318-320 by M. Fatemi et al. Thus, a direct application of eqs. (3) and (4) without considering the tilt differences between the two wafers C1 and C2 would lead to erroneous results.